Bourbaki: Universal Quantification Interpretation

Question

I'm trying to understand Bourbaki's definition of universal quantification. The definition is on Page 36 in Theory of Sets as follows:

$(\exists x R) \equiv (\tau_{x} \mid x) R$

$(\forall x R) \equiv \lnot ((\exists x) \lnot R)$

For example:

$R = \in x y$

The resulting formula is: $\lnot (\lnot \in (\tau \space \lnot \in \square y) y)$

The resulting tree is:

It's my understanding the $\square$ is a distinguished object that satisfies the evaluated truth of $\tau$. Therefore, if there is a distinguished object $\square$ that satisfies $\tau$, i.e. a object that does not satisfy $\in x y$, the universal quantification is false.

Does the quantification function $\tau$ try every object in the domain of discourse? If so, and all the objects in the domain of discourse do not satisfy $\tau$ (i.e. universal quantification should be true), what does $\tau$ evaluate to?

It seems that $\tau$ must evaluate to an object that satisfies $\in x y$ for the formula to be true, however I'm unsure how this occurs as every object in the domain of discourse must be tested first for universal quantification to be true.

Any guidance here appreciated. Thanks

Answer 1

See page 20 :

Let us consider the assertion $B$ as expressing a property of the object $X$; then, if there exists an object which has the property in question, $\tau_X(B)$ represents a distinguished object which has this property; if not, $\tau_X(B)$ represents an object about which nothing can be said.

The source is the so-called Hilbert's Epsilon Calculus :

The intended interpretation is that $ε x A$ denotes some $x$ satisfying $A$, if there is one.

Quantifiers can be defined as follows:

$$∃x A(x) \equiv A(ε x A)$$

$$∀x A(x) \equiv A(ε x (¬ A)).$$

Compare with page 36 :

$(\tau_x(R) \mid x)R$ is denoted by "there exists $x$ such that $R$".

The intuition is (quite) simple : if there are some objects $X$ such that $A$ holds of them, the "choice operator" $\tau_X$ will pick up one of them and obviously $A$ will hold of it.

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Consider now the example with the universal quantifier : $\forall x R(x)$ [where $R(x)$ is $\in (x,y)$, i.e. $(x \in y)$].

If $\forall x R(x)$ holds, this means that $\lnot R(x)$ holds of no object.

According to the above specification, $\tau_x (\lnot R)$ will pick up an object whatever, and $\lnot R(x)$ does not hold of that object, i.e. $\lnot [((\tau_x (\lnot R)) \mid x) (\lnot R)]$.

In plain language : if $\lnot R$ does not hold of an object whatever, this means that $R$ holds of every object.

Answer 2

The concept here is that by some means, $\tau_x(R)$ picks - once for all time - an otherwise completely arbitrary value of $x$ such that $R$ is true. If there is no value of $x$ for which $R$ is true, then $\tau_x(R)$ simply picks an arbitrary object.

Think of it as some munificient deity has compiled a list of all possible relations involving $x$, and for each assigned at random a value that makes it true, provided that such a value exists. Otherwise, it assigned a value completely at random. Once this list and assignments are established, $\tau_x(R)$ will always be the value assigned to $R$ in the list. Since the values assigned are random, when $R(\tau_x(R))$ is true, the only things that are knowable about $\tau_x(R)$ are the theorems that can be proved from $R(\tau_x(R))$. And when $R(\tau_x(R))$ is false, the only things knowable about $\tau_x(R)$ are things than can be proven for all values.

Of course, these are just the intuitions behind the operator. From the formalist view of Bourbaki in this book, it is really just strings of symbols being manipulated in accordance with certain rules.

But say what you will about their shortcomings (and I agree), I will still hold a fond spot for formalism in general and Bourbaki's in particular, for first demonstrating to me that the question of "what is the true mathematics" is meaningless. As long as the axioms you've chosen are consistent, your theory is just as "true" as any other. That is mathematical freedom.